Quantum synchronizable codes are kinds of quantum error-correcting codes that
can not only correct the effects of quantum noise on qubits but also the
misalignment in block synchronization. In this paper, we construct two classes
of quantum synchronizable codes which obtained by the cyclotomic classes of
order two over $mathbb{Z}_{2q}$. And the synchronization capabilities reach
the upper bound under certain constrains. Moreover, the quantum synchronizable
code posses good error-correcting capability towards bit error and phase error,
since cyclic codes we employed to construct quantum synchronizable codes are
optimal or almost optimal.

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